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Porton, Victor
Upgrading a multifuncoid ★★
Author(s): Porton
Let be a set,
be the set of filters on
ordered reverse to set-theoretic inclusion,
be the set of principal filters on
, let
be an index set. Consider the filtrator
.
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \mathfrak{P}^n $](/files/tex/81d0b4bc571faee2e8f352e99db8b54be6f9bb4f.png)
![$ E^{\ast} f $](/files/tex/93b48164cf5af924121c451b6bb17268ae140bae.png)
![$ \mathfrak{F}^n $](/files/tex/34e3ea9e5283eb80b19513453991c07af9c98f8a.png)
See below for definition of all concepts and symbols used to in this conjecture.
Refer to this Web site for the theory which I now attempt to generalize.
Keywords:
Strict inequalities for products of filters ★
Author(s): Porton
![$ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B} \subset \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} $](/files/tex/333c9bbd34eb26a9a813d527283c1b038b247af3.png)
![$ \mathcal{A} $](/files/tex/3abde4ab7e21fe6fad91d0a03ad306c2c82659d9.png)
![$ \mathcal{B} $](/files/tex/cca7b496bd14e6acf10041305acbd75cd720f9b3.png)
![$ \mathcal{A} = \mathcal{B} = \Delta \cap \uparrow^{\mathbb{R}} \left( 0 ; + \infty \right) $](/files/tex/34af23595994335e2e2b76a7cd787c6fce57e186.png)
A weaker conjecture:
![$ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B} $](/files/tex/64050fc1df5a7d5bc9cf7f4f90fe71a78d003a0b.png)
![$ \mathcal{A} $](/files/tex/3abde4ab7e21fe6fad91d0a03ad306c2c82659d9.png)
![$ \mathcal{B} $](/files/tex/cca7b496bd14e6acf10041305acbd75cd720f9b3.png)
Keywords: filter products
Join of oblique products ★★
Author(s): Porton
![$ \left( \mathcal{A} \ltimes \mathcal{B} \right) \cup \left( \mathcal{A} \rtimes \mathcal{B} \right) = \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} $](/files/tex/985106da22b932d63d15946e484e9de0ba9c261d.png)
![$ \mathcal{A} $](/files/tex/3abde4ab7e21fe6fad91d0a03ad306c2c82659d9.png)
![$ \mathcal{B} $](/files/tex/cca7b496bd14e6acf10041305acbd75cd720f9b3.png)
Keywords: filter; oblique product; reloidal product
Outer reloid of restricted funcoid ★★
Author(s): Porton
![$ ( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f) \cap^{\mathsf{RLD}} ( \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}) $](/files/tex/24a4dc2468c5502a9522738cf4ff249d4d99374b.png)
![$ \mathcal{A} $](/files/tex/3abde4ab7e21fe6fad91d0a03ad306c2c82659d9.png)
![$ \mathcal{B} $](/files/tex/cca7b496bd14e6acf10041305acbd75cd720f9b3.png)
![$ f\in\mathsf{FCD}(\mathrm{Src}\,f; \mathrm{Dst}\,f) $](/files/tex/afde5f764fb34cc3db4ef80023d22c445a0805b5.png)
Keywords: direct product of filters; outer reloid
Characterization of monovalued reloids with atomic domains ★★
Author(s): Porton
- an injective reloid;
- a restriction of a constant function
(or both).
Keywords: injective reloid; monovalued reloid
Composition of reloids expressed through atomic reloids ★★
Author(s): Porton
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
![$$g \circ f = \bigcup \{G \circ F | F \in \mathrm{atoms}\, f, G \in \mathrm{atoms}\, g \}.$$](/files/tex/f621df0db7ef2ef6d3fe1b5c523902f8b37e8217.png)
Keywords: atomic reloids
Outer reloid of direct product of filters ★★
Author(s): Porton
![$ ( \mathsf{\tmop{RLD}})_{\tmop{out}} ( \mathcal{A} \times^{\mathsf{\tmop{FCD}}} \mathcal{B}) = \mathcal{A} \times^{\mathsf{\tmop{RLD}}} \mathcal{B} $](/files/tex/167a9e622aabae06c6c07f9e889a37d3d269b0f2.png)
![$ \mathcal{A} $](/files/tex/3abde4ab7e21fe6fad91d0a03ad306c2c82659d9.png)
![$ \mathcal{B} $](/files/tex/cca7b496bd14e6acf10041305acbd75cd720f9b3.png)
Keywords: direct product of filters; outer reloid
Chain-meet-closed sets ★★
Author(s): Porton
Let is a complete lattice. I will call a filter base a nonempty subset
of
such that
.
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ \mathfrak{A} $](/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png)
![$ T\in\mathscr{P}S $](/files/tex/57e116e15e721bcdb86104a2c28f1cdfb0e6df9e.png)
![$ \bigcap T\in S $](/files/tex/8f696ee668ef8dd7ea6c8a64b9669645285fc295.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ \mathfrak{A} $](/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png)
![$ T\in\mathscr{P}S $](/files/tex/57e116e15e721bcdb86104a2c28f1cdfb0e6df9e.png)
![$ \bigcap T\in S $](/files/tex/8f696ee668ef8dd7ea6c8a64b9669645285fc295.png)
Keywords: chain; complete lattice; filter bases; filters; linear order; total order
Co-separability of filter objects ★★
Author(s): Porton
![$ a $](/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png)
![$ b $](/files/tex/b94226d9717591da8122ae1467eda72a0f35d810.png)
![$ U $](/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png)
![$ a\cap b = \{U\} $](/files/tex/d629028c6ede583e9f727914a84518040302cf11.png)
![$$\exists A\in a,B\in b: (\forall X\in a: A\subseteq X \wedge \forall Y\in b: B\subseteq Y \wedge A \cup B = U).$$](/files/tex/383c8f7cfee251bbc6606fbb6a41fbb185d817dc.png)
See here for some equivalent reformulations of this problem.
This problem (in fact, a little more general version of a problem equivalent to this problem) was solved by the problem author. See here for the solution.
Maybe this problem should be moved to "second-tier" because its solution is simple.
Keywords: filters
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